Hyperbolicity and uniformly Lipschitz affine actions on subspaces of $L^1$

TL;DR

This paper demonstrates that hyperbolic and acylindrically hyperbolic groups can act properly and unboundedly on subspaces of $L^1$ spaces via Lipschitz affine actions, using advanced geometric tools.

Contribution

It establishes new affine action results for hyperbolic groups on $L^1$ subspaces, extending understanding of their geometric group actions.

Findings

Hyperbolic groups admit proper Lipschitz affine actions on $L^1$ subspaces.

Acylindrically hyperbolic groups have unbounded Lipschitz affine actions on such spaces.

Utilizes Mineyev's $Q$-bicombings and Balasubramanya's quasi-tree actions.

Abstract

We show that every hyperbolic group has a proper uniformly Lipschitz affine action on a subspace of an $L^1$ space. We also prove that every acylindrically hyperbolic group has a uniformly Lipschitz affine action on such a space with unbounded orbits. Our main tools are the $\mathbb{Q}$-bicombings on hyperbolic groups constructed by Mineyev and the characterisation of acylindrical hyperbolicity in terms of actions on quasi-trees by Balasubramanya.